Weighted inequalities for the one-sided Hardy-Littlewood maximal functions

Abstract:

Let ${M^ + }f(x) = {\sup _{h > 0}}(1/h)\int _x^{x + h} {|f(t)| dt}$ denote the one-sided maximal function of Hardy and Littlewood. For $w(x) \geqslant 0$ on $R$ and $1 < p < \infty$, we show that ${M^ + }$ is bounded on ${L^p}(w)$ if and only if $w$ satisfies the one-sided ${A_p}$ condition: \[ \left ( {A_p^ + } \right )\qquad \left [ {\frac {1} {h}\int _{a - h}^a {w(x)dx} } \right ]{\left [ {\frac {1} {h}\int _a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right ]^{p - 1}} \leqslant C\] for all real $a$ and positive $h$. If in addition $v(x) \geqslant 0$ and $\sigma = {v^{ - 1/(p - 1)}}$,then ${M^ + }$ is bounded from ${L^p}(v)$ to ${L^p}(w)$ if and only if \[ \int _I {{{[{M^ + }({\chi _I}\sigma )]}^p}w \leqslant C\int _I {\sigma < \infty } } \] for all intervals $I = (a,b)$ such that $\int _{ - \infty }^a {w > 0}$. The corresponding weak type inequality is also characterized. Further properties of $A_p^ +$ weights, such as $A_p^ + \Rightarrow A_{p - \varepsilon }^ +$ and $A_p^ + = (A_1^ + ){(A_1^ - )^{1 - p}}$, are established.